
theorem
  for S being non void feasible ManySortedSign for S9 being non void
  Subsignature of S for A1,A2 being MSAlgebra over S st the Sorts of A1
is_transformable_to the Sorts of A2 for h being ManySortedFunction of A1,A2 st
h is_homomorphism A1,A2 ex h9 being ManySortedFunction of A1|S9, A2|S9 st h9 =
  h|the carrier of S9 & h9 is_homomorphism A1|S9, A2|S9
proof
  let S be non void feasible ManySortedSign;
  let S9 be non void Subsignature of S;
  let A1,A2 be MSAlgebra over S such that
A1: the Sorts of A1 is_transformable_to the Sorts of A2;
  set f = id the carrier of S9, g = id the carrier' of S9;
A2: f,g form_morphism_between S9,S by Def2;
  then reconsider f as Function of the carrier of S9, the carrier of S by Th9;
  let h be ManySortedFunction of A1,A2;
  assume h is_homomorphism A1,A2;
  then consider
  h9 being ManySortedFunction of A1|(S9,f,g), A2|(S9,f,g) such that
A3: h9 = h*f and
A4: h9 is_homomorphism A1|(S9,f,g), A2|(S9,f,g) by A1,A2,Th34;
  consider k being ManySortedFunction of A1|S9, A2|S9 such that
A5: k = h9 and
  k is_homomorphism A1|S9, A2|S9 by A4;
  take k;
  thus thesis by A3,A4,A5,RELAT_1:65;
end;
